A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings

For complex parameters a, c, we consider the Hénon mapping Ha,c : C → C, given by (x, y) 7→ (x + c−ay, x), and its Julia set, J . In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J , which is preserved by DH, and a continuous norm in which DH (and DH) uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is hyperbolic on J . We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.